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Strazewski Journal of Systems Chemistry 2010, 1:11 http://www.jsystchem.com/content/1/1/11

PERSPECTIVES Open Access The relationship between difference and ratio and a proposal: Equivalence of temperature and time, and the first spontaneous breaking Peter Strazewski

Abstract The mathematical form of the energy-entropy relationship is that of the relationship between the difference and the ratio of any two ‘entities’ x and y creating a geometrical 3D projection x = y · z, i.e., a surface of the shape of a hyperbolic paraboloid being a particular form of a quadric. The significance of this relationship is discussed here in a realm beyond thermodynamics. Somewhat intuitively, yet still in strict mathematical analogy to the relation between its most fundamental state variables, I propose for any isolated the cosmological state variables, absolute temperature and absolute time, to be equivalent much in the same way as energy and entropy are equivalent for an isolated ensemble of similar objects. According to this principle, the is inversely proportional to the squared product of absolute time and absolute temperature. The spontaneous symmetry breaking into a time component and a temperature component of the universe takes place when the first de facto irreversible movement occurs owing to a growing accessed total volume.

For any element aa there is an inverse element –1 : The relationship between difference and ratio in (G3) mathematics and physics aa ––1 = a11  ae= In mathematics the concept of ‘agroup’ is used to for- mulate, in a most general way, any kind of mathematical Any pair of elements ab and are commutative : operation, for example, addition, multiplication or rota- (G4) = tion. Generally, a mathematical group consists of ‘ele- ab ba ’ ments a, b, c, ... and an instruction that attributes to The mathematical operationofadditioninconjunc- each pair of elements, say a and b, a new element ab tion with elements that are integers, {a, b, c, ...} Î ℤ,is in such a way that three group axioms G1, G2 and G3 an abelian group. To specify, replace the above general strictly apply. Abelian groups are commutative for formalism with a specific one, i.e., replace  with +, e – which group axiom G4 applies as well: with 0, and a 1 with –a. The mathematical operation of For any three elements ab,: and c associativity applies multiplication in conjunction with elements that (G1) are positive real numbers, {a, b, c,...}Î R+,isalso (()ab ca= () bc − an abelian group:  :,:=⋅e =1 , and aa1 :/=1 . Note that, whereas the addition of real numbers is an abelian For any element ae there is an identity element : group, axiom G3 usually does not apply to the multipli- (G2) + ea== ae a cation of positive integers {a, b, c, ...} Î ℤ because gen- – – – erally {a 1, b 1, c 1,...}∉ ℤ. Most commonly, mathematical groups are used to describe geometrical symmetries; the vast majority of mathematical groups are non-abelian (axiom G4 does not apply), in particu- Correspondence: [email protected] lar, when the parameter space is higher than 2D (two- Laboratoire de Synthèse de Biomolécules, Institut de Chimie et Biochimie Moléculaires et Supramoléculaires (CNRS UMR 5246), Université Claude dimensional). Bernard Lyon 1, Université de Lyon, 43 bvd du 11 novembre 1918, F - 69622, Villeurbanne, France

© 2010 Strazewski; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Strazewski Journal of Systems Chemistry 2010, 1:11 Page 2 of 7 http://www.jsystchem.com/content/1/1/11

In physics and this formalism is generalised fashion the relationship between difference u extremely useful for the description of spontaneous and ratio x/y as depicted in Figure 1. Note that v is the symmetry breaking into different fundamental forces root of the function in equation 2. through the application of gauge theories, such as in To analyse more readily, a set of 2D projections of the (QED) or Quantum Chromo- same function is depicted in Figure 2 where u is dynamics (QCD) being the basis of the . sampled over a representative range of numerical values A is a type of theory in which the for x and y. Note that, trivially, x/y requires for v >0 Lagrangian is invariant under a certain continuous the root to lie in the quadrants where both x and y have group of local transformations. The Lagrangian of a thesamesign(righthalfof the 2D plot), whereas a dynamical system is a function that summarises the negative value for v places the root in the quadrants of dynamics of the system. Common to all these theories opposite signs for x and y (reflection of the hyperbles is the utilization of Legendre transformations. The through x/y = 0, not shown). Legendre transformation can be generalized to the In the relationship between difference Legendre-Fenchel transformation. It is commonly used and ratio immediately visualises how the shape of in thermodynamics and in the Hamiltonian formulation squared distance differentials in ds2 (ordinate of , for example, to describe various in the above 2D plot) relates to the ratio of squared thermodynamic potentials in classical thermodynamics, spatial distance differentials over squared time differen- 2 2 2 to derive partition functions from a state equation in sta- tials d/dxti for v =c (abscissa). In thermodynamics tistical thermodynamics, to link Lagrange (classical) the above relationship shows how changes in Gibbs free mechanics with Hamiltonian (quantum) mechanics, to energy ΔGT relate to the temperature at which ΔG formulate QED and QCD, and more. In a thermody- vanishes, T = ΔH/ΔS for ΔCp = 0, or in a differential namic context the linking of energy with entropy through form for infinitesimal state changes: dGT vs. T =dH/dS. a quantified similarity argument, as described in group It was shown that the ΔH and ΔS values of similar thermodynamics [1], may be viewed as a Legendre trans- objects at their respective equilibrium temperatures are formation on a group, rather than individual, level. all linearly related to their corresponding ΔGT values Consider a mathematically general relationship [1]. These realised experimental values of similar objects between difference and ratio (quotient) being the opera- gather linearly, in the above plot, close to the root of tional results obtained from, respectively, the substrac- the function at {x – v · y =0,x/y = v}, i.e., close to the tion and division of the same elements, say, x and y saddle point. Physics imposes positive v values in both (eqns. 1 and 2): special-relativity and thermodynamics, since there is no immediate physical meaning for a negative speed of ux=−⋅ y (1) light (reaching in vacuo a maximum constant positive value c) nor negative absolute temperature, both, in xy//()=⋅ x x − u (2) accord with the Second Law. General and special relativity describe the shape of Interpret x and y as any two parameters that are and the dynamics within a spacetime where the objects needed to define the characteristics of some state and may or may not move in a reversible fashion. The main v as a mathematically independent variable. For exam- focus in relativity theory is not reversibility but rather to ple, {x, y} may be the coordinates of a point in 2D space describe objects (particles) and their ‘directed’ move- and v a force acting on one coordinate (in one direc- ments at up to relativistic speeds along geodesics in tion) but not the other. If x were to represent the spacetime being shaped by gravitational fields. Thermo- distance between two objects in Euclidian 3D space, dynamics describes the energetics of dynamic systems difference u would describe the movement of these that contain multi-component objects (‘macrostates’) objects through spacetime according to Einstein’s special able to adopt, essentially reversibly, multiple isoergonic 2 =−⋅2 22 relativity ddcdsxi twhere d = differential, s = (degenerate) ‘microstates’. Both theories are, in terms of distance in spacetime, xi = spatial Cartesian coordinates physics, quite different from one another. The former is {x, y, z}, c = vacuum speed of light, and t = time. Differ- based on Einstein’s mass-energy equivalence; in the lat- ence u could also represent the incremental change Δ in ter an energy-entropy equivalence of macrostates was free energy upon transformation of one (metastable) identified within groups of similar objects. Mathemati- macrostate into another, as expressed in the Gibbs- cally, the former is generated from applying Pythagorean ΔΔ=−⋅ Δ Δ Helmholtz equation GHTST ,where GT = rules to the coordinates of a 4D Lorentzian geometry GibbsfreeenergychangeattemperatureT, ΔH = (two sums and a difference between four positive terms: energy change at constant pressure (enthalpy change), 2 ++−⋅2 2 22 dddcdxxx1 2 3 t). The latter originates from and ΔS = entropy change. Equation 2 describes in a correlating and partitioning energy contributions Strazewski Journal of Systems Chemistry 2010, 1:11 Page 3 of 7 http://www.jsystchem.com/content/1/1/11

Figure 1 3D view of the relationship between difference and ratio. Equation 2 at v = 300. The saddle point of this hyperbolic paraboloid is {u = x – v · y =0,x =0,x/y = 300}. through Legendre transformations. In spite of this the assumption, unproven as it is, that the energy- apparent disparity, the mathematical formalism looks entropy relationship for an ensemble of similar objects alike for both theories. Lorentzian geometry and the [1] applies to the energetics of all physical changes Gibbs-Helmholtz equation are both typified through a irrespective of scale and material; actually, irrespective crucial substraction term; through the mathematical dif- of whether it is applied to normal matter, , ference viz. the physical balance between two most fun- radiation or the fabric of spacetime itself. The basis damental components of spacetime and, respectively, stems from mathematics; more precisely, from the free energy. What is the squared spatial distance differ- relationship between difference and ratio of entities 2 ential in special relativity, dxi ,isformallytheenergy that are quantifying fundamental states of similar differential in thermodynamics, dU (or dH at constant objects in the universe. Consider the fact that the uni- pressure). The spacetime differential that describes verse, as we know it now, took its beginnings with a changes in time not space, c2 ·dt2, formally becomes the , meaning that all of a sudden its total volume entropic component of the free energy differential, T·dS. began to grow at rates that would vary. This change- The balance between each pair of components gives rise able growth rate is usually seen as varying with time; to the squared spatio-temporal distance differential ds2 here we shall prefer to see, both, growth rate and time and,respectively,thefreeenergydifferentialdF (or dG varying with the universe’stotalvolume.Itgrowsfor at constant pressure). the simple reason that it can access to more spacetime through a not yet fully understood expansion mechan- Inherently irreversible (de facto not reversed) ism being fueled by a dynamically biased balance versus essentially reversible dynamics between attractive force fields of normal and dark mat- What comes next is in part intuition rather than firm ter on the one hand, and repulsive force fields of calculation all the way through. The starting point is spacetime on the other. Strazewski Journal of Systems Chemistry 2010, 1:11 Page 4 of 7 http://www.jsystchem.com/content/1/1/11

Figure 2 2D view of the relationship between difference and ratio. Equation 2 at v = 300 (lower scale for values of x/y)orv = 1 (upper scale for values of x/y). Slices through x = –128, –64, –32, –16, –8, –4, –2, –1, 1, 2, 4, 6, 8, 16, 32, 64, 128 are shown and labelled in colour code. Datapoints were generated using y = –16 to –0.0001 and 0.0001 to 16. The orthogonal asymptotes of all hyperbles are u = x – v · y =±∞ for x/y ® 0 and u = x – v · y = x for x/y ® ± ∞. The only root of the function, thus, the saddle point formed by all hyperbles (cf. Fig. 1) is always x/y = v.

Prior to this expansion, the total of existing spacetime The universe, at any of its states, can be globally char- must have been in a minimum entropy state — or else acterised through the state variables time of cosmic the stringency of the Second Law could not apply — and expansion, call it cosmological or absolute time t,and must have accessed a minimum volume, the Planck average absolute temperature T.Whatisviewedasan volume. Deeper insight into the universe’s ‘fabric of ‘independent variable’ in thermodynamics, absolute tem- ’ could emerge from the description perature T, finds its alter ego in : the of its dynamics through the spontaneous and continuous total volume V of a universe. Thus, V becomes the replication and disappearance of a fundamental elemen- mathematically independent variable. Absolute tempera- tary unit of spacetime that could generate and harbour, ture T is seen as a measure for energy density that man- among other ‘things’, matter and that would evolve in ifests itself in reversible stochastic ‘undirected’ complexity through a Darwinian-like process [2-4]. A movements of the constituents (a generalised ‘Brownian promising approach to a theory of is the motion’), thus, is a measure for kinetic energy density of causal dynamic triangulation (CDT) of 4D Lorentzian the reversible. Integration of T over all spacetime results spacetime geometry, in which piece wise flat geometries in its total volume-dependent value TV.Inthiscontext (‘4D-triangles’ called four-simplices) are being proposed TV might be expressed in units of the cosmological – – as fundamental elementary units of spacetime that equation of state of a perfect fluid, [m2·K·s 2·J 1], thus, assemble and create “discretised curvatures” [5]. that of the squared characteristic thermal speed of Strazewski Journal of Systems Chemistry 2010, 1:11 Page 5 of 7 http://www.jsystchem.com/content/1/1/11

particles, or of elementary units of spacetime, scaled by of the universe, from its entanglement with temperature the universal gas constant R. at a given total volume TV. TV is the measure for an Cosmological absolute time t [seconds] is also viewed average energy density of the reversible ‘fraction’ of the as an energy density, yet manifests itself in the abun- universe (perhaps one should replace ‘fraction’ with dance of de facto not reversed ‘directed’ motion, hence, ‘aspect’) and is therefore globally diminishing. It makes is a measure for kinetic energy density of the irreversible. sense to formulate these two aspects as fundamental Time only exists if there is change, most generally, cosmic variables that characterise the universe’s global change in the state of the object(s) in question; objects state at every stage of its being; as ‘two aspects of the being particles or elementary units of spacetime. In a same coin’ (= universe); as synonyms to some extent. universe small enough to allow absolutely all move- This is comparable to the synonymity of mass and ments to be strictly reversible, and de facto ‘constantly’ energy where c2 is the ‘independent variable’. (As a mat- reversed, there is no way of measuring nor defining ter of fact, c is the most constant ‘variable’ that we can time. In such a situation time does not exist within the think of; other may bear other physics owing accessed volume. As soon as the first object moves in a to other numerical values for c.) It is also comparable to first unreversed direction, or the first elementary unit of the synonymity of energy and entropy for similar objects spacetime replicates itself without disappearance of the where T is the ‘independent variable’.Noticethatin replica, a de facto unreversed motion materialises due to both special relativity and thermodynamics the most the universe’s growing size. Time begins to exist (to prominent relationships are balanced by a mathema- bear a physical meaning) and grows older with the tical difference between the respective fundamental 2 =−⋅2 22 accomplished distance multiplied by the number of state variables, dsx dcdi tand, respectively, objects performing such ‘directed’ movements. ddFUTS=−⋅ d. In contrast, general relativity calls for Both state variables t and T, being measures for differ- the substantiation of a repulsive (positive) cosmological – ent aspects of energy density, are system-immanent, are constant Λ [s 2] through the linking of its most funda- not measured by an external inertial observer but are mental state variables in a reciprocal relationship as defined from within, and so could be applied in a back- shown in equation 3. ground-independent and nonperturbative theory of − / quantum gravity [3-5]. tT⋅= c 2112 ⋅ R ⋅Λ – (3)

“More and more, I have the feeling that quantum Another reciprocal time-temperature relationship, theory and general relativity are both deeply wrong tT05. ⋅ ∝ c , has been derived from special relativity by about the nature of time. It is not enough to combine George Gamow [7-10], the co-founder of the RNA Tie them. There is a deeper problem, perhaps going back Club. This relationship still serves as the basis for to the origin of physics. Around the beginning of the explaining nucleosynthesis phenomena during early seventeenth century, Descartes and Galileo both stages of the universe. More recently Isasi obtained made a most wonderful discovery: You could draw a tT0. 827 ⋅ ∝ c (derived from Table four in [11]). graph with one axis being space and the other being Owing to the isolation of any universe de definitio and time. [...] In this way, time is represented as if it were the First Law, the overall energy density remains con- another dimension of space. Motion is frozen, and a stant throughout all states of a universe. What changes, whole history of constant motion and change is pre- due to a changing total volume, is the ratio of its irre- sented to us as something static and unchanging. If I versible versus reversible energy densities: in an expand- had to guess (and guessing is what I do for a living), ing universe the former grows to the cost of the latter, this is the scene of the crime. We have to find a way time t grows older as temperature T cools down. Several to unfreeze time - to represent time without turning consequences and questions arise from this relationship, it into space”. Lee Smolin, 2006 [6] some of them may seem rather strange or naive. Let me ask them anyway. Time unfrozen Onemustallowtheuniversetodevelopinhomogen- Questions, the first spontaneous symmetry eously — at least the matter within, to some minimal breaking, and consequences extent even at its early stages. Absolute time t is there- While the entropy and similarity of objects in the universe fore expected to grow older inhomogeneously within the grows with time, the overall temperature diminishes and accessed total volume V, thus, to be best described by a gives way to structured spacetime to the cost of structured specific energy density tensor of space. Let us derive matter; ‘structured’ spacetime in the sense of ‘de facto irre- absolute time t,beingthede facto irreversible ‘fraction’ versibly formed’. Could this be the basis — seemingly Strazewski Journal of Systems Chemistry 2010, 1:11 Page 6 of 7 http://www.jsystchem.com/content/1/1/11

paradoxically — for the spontaneous non-ergodic growth Furthermore, when, like in group thermodynamics [1], of irreversibly formed ‘structure’, which may be read out the whole universe is treated as an isolated system of as ‘information’, thus, the basis for a spontaneously similar objects, an analogous relationship of difference increasing information density in spite of a growing num- between t and 1/T should bear a physical meaning, one ber of black holes (maximum entropy objects) that are that formally compares to thermodynamic free energy expected to evaporate at ever increasing rates? or to trajectories on geodesics of relativistic spacetime: − Secondly, the limiting case for the cosmic beginning, If RTt⋅⋅=⋅Λ12/ c 2 1/ [J · s · K-1 ·m-2](eqn.3),whatis −− −− that is, for a vanishing total volume, is described in c 21121⋅⋅RTsVtΛ / ⋅−⋅⋅= 3 [s]? Like for total equation 4. energy density in classical general relativity, the ratio of reciprocal temperature over time, 1/T·t, is determined −1 lim tT= (4) by a volume-independent ratio of fundamental constants → − V 0 ()R ⋅⋅Λ12/ c 2 . The above difference between reciprocal temperature and time consequently describes the Within an extremely small total volume (close to the volume-dependence of the flow of total energy density Planck volume) all movements of all particles, or the from de facto reversed into increasingly not reversed formation and vanishing of spacetime elements, are states. exactly reversible. There is no way at all for any change It follows from the analogy to group thermodynamics to manifest itself at any given (extremely small) V,and that similar cosmic objects, or similar regions of space- so time and temperature are exactly equivalent (eqn. 4). time elements, can be characterised through the correla- The spontaneous symmetry breaking — most likely the tion between the above difference and ratio and have first of all — happens when V becomes such that the the group-characteristic shape of a 5D hyperbolic para- first precise back movement is too improbable to occur. boloid, projections of which are depicted in Figures 1 At such a critical V, time becomes first manifest and 2. This shape bears an all-negative curvature and through not reversed particle movements, or not the part that is close to the saddle point is produced by reversed spacetime formation, and thus becomes distin- state variables of identical sign, thus, originates from guishable from reciprocal temperature. strictly positive physical constants including Λ (eqn. 3). Backwards in time: As V approaches the Planckian This hyperbolic paraboloid is reminiscent of the saddle- scale, the universe’s degeneracy diminishes and its tem- shaped 4D space geometry of a universe with negative perature grows at a hyperbolically increasing rate. Only energy density, as described by the anti-de-Sitter space/ a non-zero total volume prevents the (weak) singularity conformal field theory correspondence (also known as of infinite absolute temperature. The existence of a first Polyakov’s conjecture and Maldacena duality). Similar non-zero dimensional spacetime element would mean cosmic objects or similar regions of spacetime elements that a finite maximal temperature (= reversible energy are expected to reveal a group-characterising linearity density) characterises an ‘unborn’ universe in which for volume-dependent energy density (= difference) ver- time cannot be defined from within but also cannot be sus reciprocal temperature over time (= ratio). This line- zero. How does this relate to the Planck time or the arity may be best at a total volume of the chronon? Does it mean that in the beginning, viz. on a universe in which both reversible and irreversible com- Planckian scale for the total volume, there is a superpo- ponents of the considered objects (spacetime regions) sition of many finite entangled (very short) times that appear balanced within the group. decohere with the first ‘directed’ motion? If yes, how Finally, useful insights might emerge from common many? Is there a finite population (= probability density) grounds for physics seen as a difference-ratio relation- of very short times (= irreversible energy densities) of ship of fundamental constants, as shown here, and Alain opposite signs averaging to nil? Connes’ formulation of noncommutative geometries Thirdly, in the formal analogy between special relativ- [12] through a deep analysis of addition and multiplica- ityandthermodynamicsthetimecomponentof tion groups and their recently explored connection to spacetime compares to the entropy component of entropy and statistical thermodynamics [13]. free energy: cd22⋅⋅tTS∝ d. For a vanishing total The author of this ‘Perspectives’ article, an organic volume, as proposed above, temperature may be biomolecular experimental chemist with an open mind = replaced with reciprocal time: TtV →0 1/ .Itfollows for physics and cosmology, would like to know from the ∝ 2 ⋅⋅ that ddcdStttV →0 / which integrates over dt physics community (see Appendix 1), which of the ∝ 22⋅ into StV →0 c/2 .Foranon-zerominimalt,is above conclusions and statements make sense, whether this the residual entropy of an unborn universe? they may be experimentally testable and, if yes, how? Strazewski Journal of Systems Chemistry 2010, 1:11 Page 7 of 7 http://www.jsystchem.com/content/1/1/11

Appendix 1 A very preliminary version of this manuscript was pub- lished on 15th June 2009 [14]. On my private calendar I marked the 11th May 2003 as the morning of inspiration on time-temperature equivalence and volume-dependent spontaneous symmetry breaking (date unproven).

Acknowledgements Many thanks to Petr Zimak, Basel, and Günter von Kiedrowski, Bochum, for very helpful discussions on physics and cosmology, and to Stuart Kauffman, University of Calgary, for listening and for his suggestion to consider the consequences of a time-temperature equivalence on the replication of elementary units of spacetime irrespective of matter being present or absent.

Received: 1 December 2009 Accepted: 1 September 2010 Published: 1 September 2010

References 1. Zimak P, Terenzi S, Strazewski P: New concept for quantification of similarity relates entropy and energy of objects: First and Second Law entangled, group behavior of micro black holes expected. J Syst Chem 2010, 1:2. 2. Smolin L: Did the universe evolve? Class Quant Grav 1992, 9:173-191. 3. Kauffman S, Smolin L: A possible solution for the problem of time in . The Third Culture, Edge Foundation Inc 1997 [http:// edge.org/3rd_culture/smolin/smolin_p1.html]. 4. Kauffman S, Smolin L: Combinatorial dynamics and time in quantum gravity. In Towards Quantum Gravity, Lecture Notes in Physics. Edited by: Kowalski Glikman J. Springer, Berlin; 2000:541:101-129. 5. Ambjørn J, Jurkiewicz J, Loll R: The universe from scratch. [http://arxiv.org/ abs/hep-th/0509010]. 6. Smolin L: The Trouble With Physics. Mariner Books, Boston, New York 2006, 256-257. 7. Gamow G: The origin of elements and the separation of galaxies. Phys Rev 1948, 74:505-506. 8. Alpher RA, Bethe H, Gamow G: The origin of chemical elements. Phys Rev 1948, 73:803-804. 9. Gamow G: Expanding universe and the origin of elements. Phys Rev 1946, 70:572-573. 10. Gamow G: Erratum: Expanding universe and the origin of elements. Phys Rev 1947, 71:273. 11. Isasi RA: Equivalence between the empty microspace and the cosmological space. [http://vixra.org/abs/0909.0015]. 12. Connes A: . Academic Press, San Diego, CA 1994, ISBN 0-12-185860-X. 13. Connes A, Consani C: Characteristic 1, entropy and the absolute point. [http://alainconnes.org/en/downloads.php]. 14. Zimak P, Terenzi S, Strazewski P: New concept for quantification of similarity relates entropy and energy of objects: First and Second Law entangled, equivalence of temperature and time proposed. [http://arxiv. org/abs/0906.2799].

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